"I am Elsa."
Translation:Minä olen Elsa.
Well, in spoken Finnish pronouns are actually fairly common, but they do not look like "minä", "sinä" etc. but get shortened in some way, depending on the region. (Of course there are people who do use "minä", "sinä" etc. but you chance sounding pompous if you use them.) However, no one would laugh at you for using "olen", although it would usually get shortened to "oon".
The way I'd shorten the forms (capital region):
"Minä olen" -> "mä oon"
"Sinä olet" -> "sä oot"
"Hän/se on" -> "se on"
"Me olemme" -> "me ollaan"
"Te olette" -> "te ootte"
"He/ne ovat" -> "ne on"
Yep, you can drop them too. :) I'd say, without having done any research on the matter, that it's slightly more common in spoken language to include the pronouns than leave them out, but it depends a great deal on the sentence.
The only instance that comes to mind where you really ought to include the pronouns is if it's some sort of comparison, because the verbs by themselves might not be strong enough, in a way, to highlight the contrast...